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Beam Models and Quantum Systems with Linear Potentials Based on Airy Functions
(2026-04-09) José A. Gómez Hernández; Juan Toribio Milané; Francis L. Álvarez Paulino; Manuel L. Reyes-Cordero
This article provides a rigorous study of Airy functions, emphasizing their construction by power series, their relation with Bessel functions, and their canonical integral representation. We highlight their asymptotic properties and structural role in mathematical physics. To illustrate their applicability, we develop three analytical models: the Euler–Bernoulli beam under self-weight, the quantum bouncer with a linear gravitational potential, and the particle in a uniform electric field. In each case, the quantization conditions and physical scales naturally emerge from the zeros of the Airy function. These results confirm the central role of Airy functions in bridging differential equations, special functions, and applied physics.